CN101484922A - Method of simulating detailed movements of fluids using derivative particles - Google Patents
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Abstract
A method of using derivative particles to simulate the detailed movement of fluid. The present invention provides a method, which provides a new fluid simulation technique that significantly reduces the non-physical dissipation of velocity using particles and derivative information. In solving the conventional Navier-Stokes equations, the method replacing the advection part with a particle simulation. When swapping between the grid-based and particle-based simulators, the physical quantities such as the level set and velocity must be converted. A novel dissipation-suppressing conversion procedure that utilizes the derivative information stored in the particles as well as in the grid points is developed. Through several experiments, the proposed technique can reproduce the detailed movements of high-Reynolds-number fluids, such as droplets/bubbles, thin water sheets, and whirlpools. The increased accuracy in the advection, which forms the basis of the proposed technique, can also be used to produce better results in larger scale fluid simulations.
Description
Technical Field
The present invention relates to a method of simulating detailed movement of a fluid using derivative particles.
Background
1 introduction
When water interacts violently with solids, air, or the water itself, it assumes a variety of configurations, including droplets/bubbles, thin films of water, and eddies, as shown in fig. 7. Such a characteristic can occur when the fluid undergoes motion characterized by a high Reynolds number, where the Reynolds number represents the relative magnitude of the ratio of inertia to the viscosity of the fluid. Water moving at high speed is a typical high reynolds number fluid. The present invention explores a physics-based simulation of such phenomena.
Assuming that the Navier-Stokes equations correctly model the physical motion of a fluid, a plausible simulation should be able to reproduce the high Reynolds number behavior of water. Although this work primarily discusses water, it is applicable to any fluid. However, these phenomena have not been satisfactorily reproduced so far. The present invention recognizes the factors that cause this failure and presents a method that allows for a realistic simulation of high Reynolds number liquid motion.
Simulation of seemingly unrealistic high Reynolds number fluids is related to numerical dissipation. In particular, discretized modeling of the navier-stokes equations inevitably requires estimation of physical quantities at non-grid points. In most methods, such values are calculated by interpolation of values of physical quantities at grid points. However, the error introduced by this approximation acts like a non-physical viscosity or numerical dissipation. This dissipation can be reduced by using a thinner grid; however, increasing the grid resolution increases computation time and memory requirements to impractical levels. Over the past few years, a number of excellent techniques have been proposed to address this problem. Introducing particles into the euler scheme can help capture inertial motion of the fluid and increase the effective resolution of the simulation. Enright et al [2002b ] introduced a particle level set method that increases the accuracy of surface tracking by spreading particles around the interface. Losasso et al [2004] propose an octree-based multi-level fluid solver that allows finer resolution simulations in more interesting areas such as water surfaces.
Unfortunately, the above techniques do not produce high Reynolds number liquids with sufficient detail and realism. The simulated fluid exhibits a greater viscosity than in real physics; fluids often look thick/viscous and do not typically exhibit movement of small scale features in complex flows. The applicant has found that the reason why the above dissipation suppression techniques produce such artifacts is that they are not able to effectively suppress velocity dissipation, even if they significantly reduce mass dissipation.
In the present invention, the applicant introduced a new concept called derivative particle (derivative particle) and developed a fluid simulator based on the concept. The applicant utilized the non-dissipative nature of the simulated raleigh's scheme for the advection part; for fluid regions that require simulation details, applicants use particles to solve the advection step. Switching between grid-based and particle-based simulators may introduce numerical dissipation. The present invention develops a new conversion procedure that allows the reproduction of detailed fluid movements. An innovative aspect of this work is that in addition to storing the physical quantities (velocity and level set values), the derivative particles also store the spatial derivatives of those quantities, which enables a more accurate assessment of the physical quantities at non-grid/non-particle locations. The use of particles for this work differs from the particle level set approach [ Enright et al 2002b ] in that the derivative particles carry the fluid velocity as well as the level set value.
Experiments show that the proposed simulator accurately tracks the interface. More interestingly, the proposed method proved to significantly reduce non-physical damping, allowing the reproduction of macroscopic motions of small scale features that occur in real high reynolds number fluids.
The remainder of the description is organized as follows: section 2 reviews previous work; section 3 gives an overview of the simulator; section 4 introduces an octree-based Constrained Interpolation Profile (CIP) solver; section 5 introduces derivative particle models; section 6 reports our experimental results; section 7 summarizes the description.
2 previous work
Since Foster and Metaxas [ 1996; 1997] first introduced a fluid animation technique based on a full 3D navier-stokes simulation, so this approach has prevailed in the computer graphics world. Jos Stam [1999] introduced a Stable fluid solver known as a Stable fluid. The advection step of the solver is implemented using the semi-Lagrangian method [ Stanifforth and C ^ ot' e 1991], which remains stable even when large time steps are used. Since then, there has been active development of fast fluid animation techniques based on the semi-lagrange method in computer graphics. Rasmussen et al [2003] propose a technique for generating 3D large-scale animations of gases using a 2D semi-lagrange solver, and Feldman et al [2003] propose an explosion model incorporating a particle-based combustion model into a stable fluid solver. Treuille et al [ 2003; 2004, to et al, introduces an optimization technique that generates fluid flows that satisfy specified key frame constraints. Additionally, Feldman et al propose the use of a hybrid mesh combining interlaced meshes and unstructured meshes [ Feldman et al 2005a ], but also the use of deformable meshes [ Feldman et al 2005b ], for which they must modify the semi-Lagrangian method.
To simulate a fluid, in addition to a stable Navier-Stokes solver, a model for tracking the surface of the liquid is required. For this purpose, Foster and Fedkiw [2001] propose a mixed surface model that introduces mass-free particles into the level set field. This model motivated the development of a particle level set method [ Enright et al 2002b ] that can capture the dynamic motion of a fluid surface with significant accuracy. Enright et al [2005] demonstrated that this particle level set approach allows the use of a large time step in the semi-lagrange scheme. This method has been used to model the interaction between fluids and rigid bodies [ Carlson et al 2004] and between fluids and deformable shell bodies [ guillelman et al 2005] and to simulate viscoelastic fluids [ Goktekin et al 2004], sand [ Zhu and Bridson 2005], multiphase fluids [ Hong and Kim 2005], and water droplets [ Wang et al 2005 ].
Instead of grid-based methods, purely particle-based methods have also been investigated. Stam and Fiume [1995] used Smooth Particle Hydrodynamics (SPH) to model the gaseous phenomenon. In SPH, the fluid is represented by a collection of particles, and the simulator calculates their motion by calculating each term of the navier-stokes equation. M. Uller et al [2003] use SPH models to simulate fluids, and in [ M1 Uller et al 2005] they use this technique to simulate multiphase fluid interactions. Premo ˇ ze et al [2003] introduced the moving-particle semi-implicit (MPS) method to the graphical community, a technique that showed better performance than SPH in simulating incompressible motion of a fluid. However, a fundamental drawback of purely particle-based methods is their lack of surface tracking capability; if an inappropriate number of particles are used, they may produce a grainy or excessively smooth surface.
The main factor that impairs the visual realism (and physical accuracy) of the simulation results is the numerical dissipation of the velocity field. To reduce dissipation when simulating the gaseous phenomenon, Fedkiw et al [2001] used cubic interpolation. They also include an additional step called vorticity constraint (vorticity constraint) that amplifies the vorticity (curl) of the velocity field, producing a true vortex-shaped portion in the flue gas motion. A more physical-based prevention of vorticity dissipation by employing the vortex (vortex) particle method [ Selle et al 2005; park and Kim 2005 ]. This method works with the vorticity-type navier-stokes equation and solves for the (solve) advection term with the particles, which results in an efficient preservation of vorticity. However, in the case of liquids, the viscosity shows a more pronounced effect; in particular, the swirling motion is short lived and less observed. Therefore, modeling the behavior of high Reynolds number liquids requires velocity dissipation suppression techniques that work in more general cases. To reduce dissipation in liquid simulation, Song et al [2005] proposed a technique based on the CIP advection method. Their technique solves the velocity advection with third order accuracy. However, with this method, the numerical viscosity from the mesh-based advection cannot be avoided. Zhu and Bridson [2005] proposed a FLuid simulation technique based on the FLuid-Implicit-Particle (FLIP) method. The FLIP method [ Brackbill and Ruppel 1986] uses particles to solve the advection step, and does not include all other steps of the grid. Although the particle advection in this approach has no numerical dissipation, interpolation errors are introduced when passing velocity values back and forth between the grid and the particles.
Therefore, there is a long felt need for a method of simulating detailed motion of a fluid using derivative particles. The present invention is intended to solve these problems and meet the long-felt need.
Disclosure of Invention
The present invention seeks to address the disadvantages of the prior art.
It is an object of the present invention to provide a method for simulating detailed movements of a fluid using derivative particles.
It is another object of the present invention to provide a method for simulating detailed motion of a fluid using derivative particles, wherein the advection part is implemented with the lagrangian scheme.
It is a further object of the present invention to provide a method for simulating detailed movements of fluids using derivative particles, which provides a procedure for converting physical quantities at the switching of advection and non-advection parts, suppressing the intervention of unnecessary numerical dissipation.
Overview of the invention 3
In the development of multiphase fluid solvers, the applicant assumed that both air and water were incompressible. The Navier-Stokes equation for incompressible fluids can be written as
Where u is velocity, p is pressure, f is external force, μ is viscosity coefficient, and ρ is fluid density. To accurately model density and viscosity discontinuities across an interface between media, applicants employed a virtual fluid method with variable density [ Liu et al 2000; kang et al 2000; hong and Kim 2005 ]. In our solver, surface tracking is based on a level set method. Updating the level set field phi according to
Symbol distance condition
For the time integration of the Naviger-Stokes equation, applicants employ a step-by-step method [ Stam1999], which incrementally calculates (accounts for) the effect of the term in equation (1) and the mass conservation condition in equation (2). The techniques developed by the applicant are focused here on the increase of the advection part. Thus, as shown in FIG. 8, applicants have split (fractional steps) into two parts, a non-advection part and an advection part. The time integration of equation (3) involves only the advection part.
The advection part consists of a grid advection part and a particle advection part. The mesh advection part advects the velocity and level set fields computed from the non-advection part according to the current velocity field. Our method differs from pure euler simulation in that our simulator includes a particle advection part that is used to simulate the area where detail needs to be generated. To this end, applicants have introduced a large number of particles in the air-water interface.
In the mesh advection part, advection transport speed and advection transport level set are euler advection steps. The precise processing of these steps forms the basis for successful simulation of high Reynolds number behaviour. For euler advection, the applicant developed an octree-based CIP solver (part 4) that reduces the speed and mass dissipation to a significant level.
When the advection of a fluid region needs to be simulated using the particle advection part, the physical quantities (e.g., level set and velocity) currently defined on the grid are delivered to the particles. After this transport, the particles can be transported in parallel flow directly. The results of the particle advection are then transmitted back to the grid. At this point in the program, the final velocity and level set values are stored at the grid points, regardless of whether the simulation was conducted via the grid advection part or the particle advection part. Applicants describe the grid-to-particle and particle-to-grid speed/level set conversion procedures in section 5. The use of particle-based advection and the development of conversion programs are essential to reproduce the details of fluid motion.
A method of simulating detailed movement of a fluid using derivative particles comprising the steps of: a) modeling the fluid with an adaptive mesh based on an octree data structure; b) solving the Navier-Stokes equations for the incompressible fluid for the fluid velocities at the grid points; c) using a step-wise method in the time integration of the Navier-Stokes equation; d) dividing the steps into a non-advection part and an advection part; e) selecting a grid advection part or a particle advection part according to the detail degree of the simulation; f) using octree-based CIP method for the mesh advection part; g) derivative particle models are used for the particle advection part.
When the level of detail of the simulation is high, the step of using the derivative particle model for the particle advection part is used.
Adapting the mesh to the details of the region of high interest increases the number of meshes.
The octree data structure comprises a tree data structure with up to eight children per internal node.
The Navier-Stokes equation for incompressible fluids can be written as
Where u is velocity, p is pressure, f is external force, μ is viscosity coefficient, and ρ is fluid density.
The method may further comprise the step of tracking the surface of the fluid using a level set method. The level set method includes a level set field phi. Updating the level set field φ according to:
distance of symbol <math> <mrow> <mo>|</mo> <mo>▿</mo> <mi>φ</mi> <mo>|</mo> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow></math>
The grid advection part advects the delivery speed and level set using an Euler advection step. Euler advection is performed by the octree-based CIP (constrained interpolation profile) method.
The step of using the octree-based CIP method for the grid advection part comprises the step of advecting the velocity and level sets and their derivatives with the octree-based CIP method. Direct from equation <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mrow></math>
Obtaining an equation for the advection transport derivative, where φ is the physical quantity advected, and obtaining an advection transport equation for the derivative of the spatial variable x by differentiating the equation for x, i.e.:
the step of solving the advection equation for the derivative of x comprises the step of using a step-by-step method, which comprises the steps of: a) solving for non-advective parts using finite difference method <math> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>/</mo> <mo>∂</mo> <mi>t</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>;</mo> </mrow></math>
b) Advective conveying device <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow></math>
The result of (1); and c) sampling pressure values at the center of cells (cells) defined by the grid and sampling other values including velocity, level sets and their derivatives at nodes.
The method may further comprise the steps of: a) combining an octree-based CIP method with an adaptive grid; and b) minimizing octree artifacts caused by regional variations in grid resolution of the octree data structure by simulating advection with the octree-based CIP method.
The derivative particle model uses a particle-based lagrangian scheme. By using <math> <mrow> <mi>φ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> </mrow> <mrow> <mo>|</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mo>|</mo> </mrow> </mfrac> <mo>·</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> </mrow></math>
And <math> <mrow> <mo>▿</mo> <mi>φ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> </mrow></math>
the level set and its derivative at the grid points are computed, where p is the gradient carried by the derivative particles.
The method may further comprise the steps of: a) converting the defined velocity and level sets on the grid to defined velocity and level sets on the particles (grid to particles) prior to the step of using the derivative particle model for the particle advection part; and b) converting the defined velocity and level sets on the particles to defined velocity and level sets on the grid (particle to grid) before returning to the step of using the octree-based CIP model for the grid advection part.
The step of converting the velocity and level set grid to particles comprises the step of, for velocity uPUsing the steps of the FLIP method with monotonic CIP interpolation replacing linear interpolation, and <math> <mrow> <msup> <mi>u</mi> <mi>P</mi> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mo>-</mo> <mi>P</mi> </msubsup> <mo>+</mo> <msubsup> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <msub> <mi>w</mi> <mi>i</mi> </msub> <mi>Δ</mi> <msubsup> <mi>u</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>,</mo> </mrow></math>
wherein
Is the particle velocity just before the start of the non-advection part, and
is due to a non-advective part of GiThe speed of (c) is changed.
For speed uPGrid points G, nearest particle P selected from each quadrant of G1、P2、P3And P4、PiVelocity of and
and
the step of converting the velocity and level set particles to a grid comprises the steps of: a) will be provided with
Derivative of (2)Coordinate transformation to
Thereby to obtainIndicates along the direction
Component (a) ofIs shown perpendicular to
Wherein similarly obtained
Derivative of the coordinate transformation of
b) Along the directionPerforming one-dimensional monotonic CIP interpolation on the result obtained in step (a) to calculate u at position AAAnd it
A directional derivative M; c) by applying the inverse (inverse) of the above-mentioned coordinate transformation
Obtaining (u)Ax，uAy) (ii) a d) To particle P3And P4Performing steps (a) - (c) to calculate u of BBAnd (u)Bx，uBy) (ii) a And e) performing a y-direction monotonic CIP interpolation on the results obtained in steps (c) and (d) which gives the velocity and derivative at G.
The step of using a derivative particle model for the particle advection part comprises the steps of: a) adjusting a level set value of the particle; and b) performing particle reseeding.
The advantages of the present invention are (1) the method of simulating detailed motion of a fluid using derivative particles reduces dissipation of both mass and velocity, resulting in a true reproduction of the dynamic fluid; (2) the method of simulating detailed movement of a fluid using derivative particles implements the advection part with the lagrangian scheme; and (3) the method of simulating detailed motion of a fluid using derivative particles provides a particle-to-grid and grid-to-particle procedure.
While the present invention has been briefly summarized, a more complete understanding of the invention can be obtained from the following drawings, detailed description, and claims.
Drawings
These and other features, aspects, and advantages of the present invention will become better understood with reference to the accompanying drawings.
FIG. 1 is a flow chart illustrating a method of simulating detailed movement of a fluid using derivative particles in accordance with the present invention;
FIG. 2 is another flow chart illustrating a method according to the present invention;
FIG. 3 is a further flow chart illustrating a method according to the present invention;
FIG. 4 is a partial flow diagram showing grid-to-particle and particle-to-grid conversion;
FIG. 5 is another partial flow diagram showing grid-to-particle and particle-to-grid conversion;
FIG. 6 is a partial flow diagram showing details of a step of using a derivative particle model;
FIG. 7 shows simulation results of water produced by the impact of a ball;
FIG. 8 illustrates the architecture of a method according to the invention;
FIG. 9 shows the calculation from particle velocity to grid velocity;
FIG. 10 shows a screenshot taken from a 2D dam break (breaking-dam) simulation: the upper and lower sequences show the results produced with a conventional linear model and derivative particle model of the particle level set method, respectively;
FIG. 11 shows a screenshot taken from a water drop simulation: the left and right images show the results produced with a conventional linear model and derivative particle model of the particle level set method, respectively;
FIG. 12 shows a screenshot taken from a simulation of a rotating water tank; and
FIG. 13 shows a screenshot taken from a flood simulation: the three images of the bottom row show the progression of the flood (top view); the top image shows a side view.
Detailed Description
4 development octree-based CIP solver
This section describes the development of the mesh advection part of the simulator by combining the CIP method with the octree data structure.
4.1 CIP method introduction
Advection terms in solving the Navell-Stokes equations (part 3) in a step-wise manner
And
special attention is required due to its hyperbolic nature. The semi-Lagrangian method provides a stable solution in which to handle the above hyperbolic equation [ Stam1999]]. Unfortunately, however, this approach suffers from severe numerical dissipation resulting from linear interpolation of the physical quantities at non-grid points used to determine a back-pull (backtracking) from the physical quantities at adjacent grid points.
While the CIP process was originally developed by Yabe and Aoki [1991 a; 1991b ] A third order technique was proposed and subsequently improved by Yabe et al [2001 ]. The key idea of this method is to advect not only its physical quantity but also its derivative. Here, the problem would be how to advect the derivative. An interesting observation of Yabe and Aoki [1991a ] is that the equation for advective transport derivatives can be obtained directly from the initial hyperbolic equation
Where φ is the physical quantity being advected. For the spatial variable x differential equation (5), applicants have obtained
Which can be used to predict the evolution of x over time. In solving equation (6), applicants again apply the step-wise approach: first, applicants use finite differences to solve for non-advection parts <math> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>/</mo> <mo>∂</mo> <mi>t</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>;</mo> </mrow></math>
Applicant then advected the results according to the following equation
A detailed description of CIP advection can be found in Song et al 2005. An extension of the two-dimensional and three-dimensional aspects of the above methods exists [ Yabe and Aoki 1991b ]. Song et al [2005] found that the generalization given in [1991b ] may cause instability and proposed a modification to address this problem. In this work, to solve the advection part, applicants employed the CIP method with a second order Runge-Kutta time integral.
4.2 combining CIP with adaptive octree Trees
The use of an adaptive mesh based on an octree data structure introduced by Losasso et al [2004] allows simulations to be performed with non-homogeneous accuracy in all fluids. From a practical point of view, this approach is useful because it allows simulating more interesting areas, such as surfaces, with higher accuracy by introducing a small amount of additional computation and memory. Here, applicants take this octree data structure and propose that the actual values of this adaptive approach can be further improved by combining it with the CIP approach.
Losasso et al [2004] use linear interpolation for the half lagrange advection step. However, if applicants replace linear interpolation with CIP interpolation, then the advection will have third order accuracy. As in Guendelman et al [2005], applicants sampled pressure values at the cell center, but sampled all other quantities-velocity, level set, and derivatives thereof-at the junction. When the cell is refined, CIP interpolation is performed on the values at the new grid points by referring to all the derivative values.
We note that the CIP method fits very well with the octree data structure because, although it has third-order spatial precision, it is confined to a single grid cell template (Tencel) rather than multiple templates. Due to this compactness, applicants are able to use the CIP method for simulating an adaptive mesh without any major modifications. In contrast, for higher order schemes defined on multiple templates, it is difficult to extend the method from first order to third order: since the simulation adaptively refines the mesh, cells with different sizes will be generated, which makes the development of multi-template high-order schemes a difficult problem.
We will also note a feature that applicant calls for octree artifacts. Regional variations in the grid resolution of the octree data structure produce variations in the amount of dissipation. The regional variation in mass dissipation is not significant. However, for velocity dissipation, regional differences can be significant, especially for fast fluid motion. For example, in the simulation of a dam break, water undergoes rapid lateral movement. Fluids near high resolution surfaces undergo a small amount of numerical diffusion and thus undergo rapid movement. Instead, the fluid at the bottom moves in a pattern characteristic of a more viscous fluid due to the low mesh resolution. When these two types of motion occur together, the upper portion of the water appears to crawl over the lower portion. High-order schemes, including CIP, do not avoid such artifacts. Then, when advection is simulated using the octree-based CIP solver proposed in this work, the artifacts are less noticeable; applicants attribute this improvement to an overall reduction in numerical dissipation, especially in the low resolution region.
5 derivative particle model
The particle-based Lagrangian scheme is more accurate than the mesh-based Euler scheme in terms of momentum/mass conservation. Therefore, the applicant applies the lagrangian scheme to regions that need to be simulated in more detail. However, this method has its own limitations in surface tracking and pressure/viscosity calculations. Therefore, the applicant applies this method only to the simulation of the advection part; all other parts of the simulation are based on the euler scheme. Switching between the lagrange and euler schemes requires grid-to-particle and particle-to-grid conversion of physical quantities (velocity and level set values). The procedures for performing these conversions should be carefully developed so that they do not introduce unnecessary numerical diffusion. In this section, applicants introduced a novel conversion procedure that does not impair the desirable properties of Lagrangian advection, which is a necessary component to enable the reproduction of small scale features in high Reynolds number fluids. For simplicity, the description in this section is made for 2D.
5.1 mesh to particle velocity conversion
At the end of the advection part shown in fig. 8, (1) the velocities that need to be advected are stored at the grid points, and (2) a large number of particles are scattered over the grid. The particle advection part must find the velocity of the particles so that their velocity and level set values produce lagrangian motion. Suppose particle P is at four grid points G1、G2、G3And G4In defined units. For i ∈ {1, 2, 3, 4}, let uG i＝(uG i，vG i) Is GiThe grid speed of (d). The applicant is looking for a speed u that can be used for PP＝(uP，vP) The formula (2).
In-cell mass point method [ Harlow 1963]One possible method used in (1) is to use bilinear interpolation <math> <mrow> <msup> <mi>u</mi> <mi>P</mi> </msup> <mo>=</mo> <msubsup> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <msub> <mi>w</mi> <mi>i</mi> </msub> <msubsup> <mi>u</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>,</mo> </mrow></math>
Wherein, wiIs a bilinear weighting determined based on the particle position P. Unfortunately, this conversion introduces severe numerical diffusion. In the FLIP method [ Brackbill and Ruppel 1986]Only the increasing portion of the grid velocity is transmitted to the site velocity. That is to say that the first and second electrodes, <math> <mrow> <msup> <mi>u</mi> <mi>P</mi> </msup> <mo>=</mo> <msubsup> <mrow> <msubsup> <mi>u</mi> <mo>-</mo> <mi>P</mi> </msubsup> <mo>+</mo> <mi>Σ</mi> </mrow> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <msub> <mi>w</mi> <mi>i</mi> </msub> <msubsup> <mi>Δu</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>,</mo> </mrow></math>
wherein,is the particle velocity just before the start of the non-advection part, and
is due to a non-advective part of GiThe speed of (c) is changed. This approach has been shown to significantly reduce numerical diffusion [ Zhu and Bridson 2005]. Thus, in the development of the grid-to-particle-velocity conversion program, applicants used the FLIP method, but interpolated with monotonic CIP [ Song et al 2005]Instead of linear interpolation. This gives not only uPAnd give
5.2 particle-to-grid speed conversion
Suppose each particle is based on velocity uPMovement, the spatial derivative of the velocity carried, and the velocity itself. Applicants must now develop a program that communicates the results of particle advection to the grid. Particle-to-grid speed conversion versus grid-to-particle conversionAnd is more complex.
Referring to fig. 9, let G be a grid point. Applicant selects the closest particle, P, from each quadrant of G1、P2、P3And P4. Make itIs PiAnd is at a speed of
And
are respectively as
The derivatives of the x and y components of (a). The applicant has to find the speed u at GG＝(uG，vG) And the formula of its spatial derivatives. Since the four particles are not placed rectangularly, applicants cannot use traditional grid-based CIP interpolation.
Our particle-to-grid velocity conversion consists of the following steps. For simplicity, applicants show the calculation of only the x component.
(1) The applicant will
Derivative of (2)
Coordinate transformation toAnd, thus,
is shown in the directionAnd u is1⊥I denotes perpendicular to
The component (c). As such, applicants have obtained
Derivative of the coordinate transformation of
(2) Applicant follows the direction
Performing one-dimensional monotonic CIP interpolation on the result obtained in step (1) to compute u at position AAAnd it
Derivative of direction
(3) Applicant derives from the inverse of the above coordinate transformation
Obtaining (u)Ax，uAy)。
(4) Similarly, Applicant pairs particle P3And P4Performing steps (1) - (3) to calculate u of BBAnd (u)Bx，uBy)。
(5) Applicants perform y-direction monotonic CIP interpolation on the results obtained in steps (3) and (4), which gives the velocity and derivative at G.
This monotonic interpolation method can be extended directly to the 3D case. Applicants have noted that this work does not employ the general radial basis interpolation scheme because (1) they do not utilize derivative information, and (2) it has proven difficult to modify the scheme so that they embody the derivatives but remain monotonous.
5.3 level set conversion
Level set conversion should be performed differently than speed conversion; the level set value represents the shortest distance to the surface, so the conversion will not be based on interpolation. Here, applicants use a widely used level set conversion program [ Enright et al 2002 a; enright et al 2002b ].
In the initial particle level set method, the spherical implicit function phi (x) is used as sp(|φp|-|x-xp| where p is the level set of particles and s is for positive (negative) particles, to compute the level set value at grid point xpAnd (1) (+ 1. In this work, using the derivative information stored in the derivative particles, applicants used the following equation to calculate the level set and its derivatives at the grid points
Where p is the gradient carried by the derivative particle. Since equation (8) is based on the distance in the gradient direction rather than the euclidean distance, it produces a more accurate result than the spherical implicit function. The conversion procedure is the same as that set forth in Enright et al [2002a ], except for the above modifications.
6 test results
The technique proposed in the present invention is implemented on a Power Mac with dual G52.5 GHz processors and 5.5GB of memory. The applicant used a program to simulate a number of experimental situations in the real world that produce high reynolds number fluid behaviour. In experiments, applicants solved a two-phase fluid consisting of water and air with the following constants: g is 9.8m/sec2、ρwater＝1000kg/m3、μwater＝1.137×103kg/ms、ρair＝1.226kg/m3And muair＝1.78×105kg/ms, where g is gravity. Extraction of water surface using marching cubes algorithm, and by smart shooting
Rendering is performed.
Dam break: to compare the numerical diffusion between the derivative particle model and the linear model using the particle level set method, applicants used an effective resolution of 1282To perform a 2D dam break test in which a column of water of 0.2 x 0.4m is released in the gravitational field. The results are shown in FIG. 10, where applicants can see that derivative particles produce less diffusion: the wave breaks more sharply and the vorticity is preserved well.
Bulk water falling onto shallow water: FIG. 11 shows a screenshot taken from the following simulation: i.e. where the bulk of the water falls onto a reservoir 0.05m deep. Applicant uses a no-slip boundary condition for the bottom surface of the reservoir which causes the water to move quickly at the top and slowly at the bottom, resulting in a crown-like splash. 256 for this experiment3Is performed at the effective grid resolution. The same scenario is also simulated with a traditional linear model using the particle level set approach. The comparison demonstrates that the proposed technique yields more realistic results. The derivative particle model uses 60 seconds per time step on average, and the linear model uses 60 seconds per time step on averageAnd 30 seconds each including the file output time.
Impact of solid ball: in this experiment, solid spheres with a radius of 0.15m were impacted into water at a velocity (5.0, -3.0, 0.0) m/sec. This impact produces the complex structure shown in fig. 7. This experiment shows that the proposed technique is able to produce detailed fluid motion and surface features that occur in violent solid-water-air interactions. The effective resolution of this experiment was 192 × 128 × 128.
Rotating the water tank: figure 12 shows a 0.9 x 0.3m spin tank with half full of water. In the experiment, the centrifugal force produced the effect of pushing the water to the tank side. This simulation is performed with an effective grid resolution of 96 x 32. This experiment demonstrates that derivative particle models can simulate complex motion of a fluid even in a coarse resolution grid.
Flood: in this experiment, applicants simulated mesoscale flooding that occurred over a 6m × 2m × 4m domain. Fig. 13 shows a screenshot taken from this experiment. When the dike is removed, the water begins to slide off the steps. The water undergoes violent movements that produce complex geometries, due to obstructions along the path. The limited resolution of this experiment was 384 × 128 × 256.
FIGS. 1-6 show flow charts of the present invention.
1-6, a method for simulating detailed motion of a fluid using derivative particles includes the steps of: a) modeling the fluid with an adaptive mesh based on an octree data structure (S100); b) solving the navier-stokes equation for the incompressible fluid for the fluid velocity at the grid points (S200); c) using a step-by-step method in the time integration of the Navier-Stokes equation (S300); d) dividing the steps into a non-advection part and an advection part (S400); e) selecting a mesh advection part or a particle advection part according to the degree of detail of the simulation (S500); f) using an octree-based CIP method for the mesh advection part (S600); and g) using the derivative particle model for the particle advection part (S700).
The step of using the derivative particle model for the particle advection part (S700) is used when the degree of detail of the simulation is high.
The adaptive mesh increases the number of meshes for the details of the region of high interest.
The octree data structure comprises a tree data structure with up to eight children per internal node.
The Navier-Stokes equation for incompressible fluids can be written as
Where u is velocity, p is pressure, f is external force, μ is viscosity coefficient, and ρ is fluid density.
The method may further include the step of tracking the surface of the fluid using a level set method (S800).
As shown in fig. 1 to 3, the level set method includes a level set field phi. Updating the level set field φ according to:
distance of symbol <math> <mrow> <mo>|</mo> <mo>▿</mo> <mi>φ</mi> <mo>|</mo> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow></math>
The grid advection part advects the delivery speed and level set using an Euler advection step. The euler advection is performed by an octree-based CIP (constrained interpolation profile) method.
The step of using the octree-based CIP method for the grid advection part (S600) includes the step of advecting the transport speed and level sets and their derivatives with the octree-based CIP method (S610). Direct from equation <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mrow></math>
Obtaining an equation for advection transport derivatives, where φ is a physical quantity advected, and obtaining an advection transport equation for a derivative of a spatial variable x by differentiating the equation for x, i.e.:
as shown in fig. 2 and 3, the step of solving the advection equation of the derivative of x (S610) includes employing a step using a step method, and the step using the step method includes the steps of: a) solving for non-advective parts using finite difference method <math> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>/</mo> <mo>∂</mo> <mi>t</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> </mrow></math>
(S620); b) advective conveying device <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow></math>
The result of (S630); and c) sampling pressure values at the centers of the cells defined by the grid and sampling other values including velocity, level sets and their derivatives at the nodes (S640).
As shown in fig. 3, the method may further comprise the steps of: a) combining an octree-based CIP method with an adaptive mesh (S650); and b) minimizing octree artifacts caused by regional variations in grid resolution of the octree data structure by simulating advection with the octree-based CIP method (S660).
The derivative particle model uses a particle-based lagrangian scheme. By using <math> <mrow> <mi>φ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> </mrow> <mrow> <mo>|</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> <mo>|</mo> </mrow> </mfrac> <mo>·</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> </mrow></math>
And <math> <mrow> <mo>▿</mo> <mi>φ</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>p</mi> </msub> </mrow></math>
to calculate the grid pointsWhere p is the gradient carried by the derivative particle, and its derivative.
The method may further comprise the steps of: a) converting the set of velocities and levels defined on the grid to a set of velocities and levels defined on the particles (S690) (grid to particles) prior to the step of using the derivative particle model for the particle advection part (S700); and b) converting the velocity and level set defined on the particles to a velocity and level set defined on the grid (S710) (particle to grid) before returning to the step of using octree-based CIP method for the grid advection part (S600).
Step of converting the velocity and level sets (S690) grid to particle includes using the FLIP method with monotonic CIP interpolation instead of linear interpolation for velocity <math> <mrow> <msup> <mi>u</mi> <mi>P</mi> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mo>-</mo> <mi>P</mi> </msubsup> <mo>+</mo> <msubsup> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <msub> <mi>w</mi> <mi>i</mi> </msub> <mi>Δ</mi> <msubsup> <mi>u</mi> <mi>i</mi> <mi>G</mi> </msubsup> </mrow></math>
And a step of conversion of uP (S692), wherein,
is the particle velocity just before the start of the non-advection part, and
is the velocity change at Gi due to the non-advective part.
As shown in fig. 5, for speed uPGrid points G, nearest particle P selected from each quadrant of G1、P2、P3And P4、PiVelocity of, and
and
the step of converting (S710) the velocity and level sets to a grid includes the steps of: a) will be provided withDerivative of (2)
Coordinate transformation to
Thus, it is possible to provide
Indicates along the direction
Component (a) of1⊥Is shown perpendicular to
Wherein similarly obtained
Derivative of the coordinate transformation of
(S712); b) along the direction
Performing one-dimensional monotonic CIP interpolation on the result obtained in step (a) to calculate u at position AAAnd it
Derivative of direction
(S714); c) by applying the inverse of the above coordinate transformation
Obtaining (u)Ax，uAy) (S716); d) to particle P3And P4Performing steps (a) - (c) to calculate u of BBAnd (a)uBx，uBy) (S718); and e) performing y-direction monotonic CIP interpolation on the results obtained in steps (c) and (d), which gives the velocity and derivative at G (S720).
As shown in FIG. 6, the step of using the derivative particle model for the particle advection part comprises the steps of: a) adjusting a level set value of the particle (S702); and b) performing particle reseeding (S704).
7 conclusion
In the present invention, applicants have introduced a new concept called derivative particles and proposed fluid simulation techniques that increase the accuracy of the advection part. The non-advection part of the simulator is implemented using the euler scheme, while the advection part is implemented using the lagrangian scheme. An important issue in developing a simulator that incorporates in this way is how to convert the physical quantities at the switching of the advection and non-advection parts. Applicants have successfully overcome this problem by developing conversion procedures that effectively suppress the unwanted dissipation of values. In fluid regions where particle advection need not be performed, advection was simulated using octree-based CIP solvers, which applicants developed in this work by combining CIP interpolation with an octree data structure. This method also serves to reduce numerical dissipation. The results of multiple experiments demonstrate that the proposed technique significantly reduces velocity dissipation, resulting in a true reproduction of the dynamic fluid.
This work would not be possible without the pioneering work of previous researchers. The idea of using particles entirely to solve the advection part comes from PIC [ Harlow 1963] and FLIP [ Brackbill and Ruppel 1986] methods, but for particle-to-grid and grid-to-particle velocity scaling, applicant uses derivative-based cubic interpolation instead of linear interpolation. In addition, the idea of letting the particles carry a level set value comes from the particle level set method [ Enright et al 2002b ]. This model is extended in this work so that the particles also carry velocity information. In addition, derivative information is used from CIP methods [ Yabe et al 2001 ]. Here, the applicant extended the application of this method to particles, leading to the development of a particle-to-grid-speed conversion program that contains extensions to the CIP method itself: CIP interpolation of particles in a non-rectangular configuration.
In the experiments, the ability to reproduce details was emphasized. However, the proposed technique can also be applied to large-scale fluid movements to produce accurate results. This technique involves an increased amount of memory for storing derivative information. However, applicants have demonstrated that 2563 grids can be simulated on current PCs, which is already a useful resolution for delineating fluid scenarios of interest.
While the invention has been shown and described with reference to various embodiments, those skilled in the art will recognize that changes may be made in form, detail, composition and operation without departing from the spirit and scope of the invention, as defined by the claims.
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Claims (19)
1. A method of simulating detailed movement of a fluid using derivative particles, comprising the steps of:
a) modeling the fluid with an adaptive mesh based on an octree data structure;
b) solving a Navier-Stokes equation for the incompressible fluid for the fluid velocity at the grid points;
c) using a step-wise method in the time integration of the Navier-Stokes equation;
d) dividing each step into a non-advection part and an advection part;
e) selecting a grid advection part or a particle advection part according to the detail degree of the simulation;
f) using an octree-based CIP method for the mesh advection part; and
g) a derivative particle model is used for the particle advection part,
wherein, when the degree of detail of the simulation is high, the step of using the derivative particle model for the mass point advection part is used.
2. The method of claim 1, wherein the adaptive mesh increases the number of meshes for regions of high interest in detail.
3. The method of claim 1, wherein the octree data structure comprises a tree data structure, wherein each internal node has up to eight children.
4. The method of claim 1, wherein the Navier-Stokes equation for incompressible fluids is written as
Where u is velocity, p is pressure, f is external force, μ is viscosity coefficient, and ρ is fluid density.
5. The method of claim 1, further comprising the step of tracking the surface of the fluid using a level set method, wherein the level set method comprises a level set field φ.
6. The method of claim 5, wherein the level set field φ | is updated according to
With signed distance <math> <mrow> <mrow> <mo>|</mo> <mo>▿</mo> <mi>φ</mi> <mo>|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow></math>
7. The method of claim 6, wherein the <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mrow></math>
The time integral of (a) involves only the advection part.
8. The method of claim 6, wherein the mesh advection portion advects the velocity and level set fields computed from the non-advection portion according to a current velocity field.
9. The method of claim 6, wherein the grid advection part advects delivery speed and level sets using an Euler advection step.
10. The method according to claim 9, wherein the euler advection is performed by octree-based CIP (constrained interpolation profile) method.
11. The method of claim 1, wherein the step of using octree-based CIP method for the advection part of the grid comprises the step of advecting the set of velocities and levels and their derivatives with the octree-based CIP method.
12. The method of claim 11, wherein the equation is directly derived from <math> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mrow></math>
Obtaining an equation for advection transport derivatives, where φ is a physical quantity advected, and obtaining an advection transport equation for a derivative of a spatial variable x by differentiating the equation with respect to x, i.e.:
13. the method of claim 12, wherein the step of solving the advection equation for the derivative of x comprises the step of using a step method, wherein the step of using a step method comprises the steps of:
a) solving for non-advective parts using finite difference method <math> <mrow> <msub> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mi>x</mi> </msub> <mo>/</mo> <mo>∂</mo> <mi>t</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>·</mo> <mo>▿</mo> <mi>φ</mi> <mo>;</mo> </mrow></math>
b) Advective conveying device <math> <mrow> <mfrac> <msub> <mrow> <mo>∂</mo> <mi>φ</mi> </mrow> <mi>x</mi> </msub> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>▿</mo> <msub> <mi>φ</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow></math>
The result of (1); and
c) the pressure values are sampled at the center of the cells defined by the grid and other values including velocity, level sets and their derivatives are sampled at each node.
14. The method of claim 1, further comprising the step of:
a) combining an octree-based CIP method with an adaptive grid; and
b) OctreeImage artifacts caused by regional variations in the grid resolution of the octree data structure are minimized by simulating advection with an octree-based CIP method.
15. The method as in claim 1, wherein the derivative particle model uses a particle-based Lagrangian scheme, wherein the level set at grid points and their derivatives are calculated with the following equation
And
where p is the gradient carried by the derivative particle.
16. The method of claim 1, further comprising the steps of:
a) converting the set of velocities and levels defined on each grid to a set of velocities and levels defined on each particle (grid to particles) prior to said step of using the derivative particle model for the particle advection part; and
b) the velocity and level sets defined on each particle are converted to the velocity and level sets defined on each grid (particle to grid) before returning to the step of using the octree-based CIP method on the grid advection part.
17. The method of claim 16, wherein the step of grid to particle conversion speed and level set comprises for speed uPUsing the steps of the FLIP method with monotonic CIP interpolation instead of linear interpolation, wherein <math> <mrow> <msup> <mi>u</mi> <mi>P</mi> </msup> <mo>=</mo> <msubsup> <mi>u</mi> <mo>-</mo> <mi>P</mi> </msubsup> <mo>+</mo> <msubsup> <mi>Σ</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <msub> <mi>w</mi> <mi>i</mi> </msub> <msubsup> <mi>Δu</mi> <mi>i</mi> <mi>G</mi> </msubsup> <mo>,</mo> </mrow></math>
WhereinIs the particle velocity just before the onset of the non-advection part, and
is due to a non-advective part of GiThe speed of (c) is changed.
18. The method of claim 16, wherein for speed u, uPGrid points G, nearest particle P selected from each quadrant of G1、P2、P3And P4The speed of Pi, and
and
the derivatives of the x-component and the y-component of (c),
the step of converting the particle to grid speed and level set comprises the steps of:
a) will be provided with
Derivative of (2)
Coordinate transformation into (u)1‖u1⊥) So that u is1‖Indicates along the direction
Component (a) of1⊥Is shown perpendicular to
Wherein similarly obtained
Derivative (u) of the coordinate transformation of (c)2‖，u2⊥)；
b) Along the direction
Performing one-dimensional monotonic CIP interpolation on the result obtained in step (a) to calculate u at position AAAnd it
Derivative of direction (u)A‖，uA⊥)；
c) By applying the inverse of the above-described coordinate transformation,from (u)A‖，uA⊥) Obtaining (u)Ax，uAy)；
d) To particle P3And P4Performing steps (a) - (c) to calculate u of BBAnd (u)Bx，uBy) B is the grid line and the connection P3And P4The intersection points between the line segments of (a); and
e) performing y-direction monotonic CIP interpolation on the results obtained in steps (c) and (d), giving velocity and derivatives at G.
19. The method of claim 15, wherein the step of using a derivative particle model for the mass advection part comprises the steps of:
a) adjusting a level set value of the particle; and
b) particle reseeding is performed.
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| US (1) | US7565276B2 (en) |
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| EP2008250A4 (en) | 2010-04-21 |
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| US7565276B2 (en) | 2009-07-21 |
| WO2007114548A1 (en) | 2007-10-11 |
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